## Abstract

Polarization-dependent photon switch is one of the most important ingredients in building future large-scale all-optical quantum network. We present a scheme for a single-photon switch in a one-dimensional coupled-resonator waveguide, where *N _{a}* Λ-type three-level atoms are individually embedded in each of the resonator. By tuning the interaction between atom and field, we show that an initial incident photon with a certain polarization can be transformed into its orthogonal polarization state. Finally, we use the fidelity as a figure of merit and numerically evaluate the performance of our photon switch scheme in varieties of system parameters, such as number of atoms, energy detuning and dipole couplings.

© 2013 OSA

## 1. Introduction

Photon, the flying qubit, is regarded as a good information carrier to implement quantum communication and large-scale quantum network [1]. As the state-of-the-art technologies develop, Fock states as a resource are available in the context of cavity and circuit quantum electrodynamics (QED) [2,3]. More particularly, high-efficiency single-photon source [4] has been realized in the system of natural and artificial atoms [5, 6]. To manipulate the photons efficiently, various photonic devices in the quantum network with atoms, such as quantum memories [7], beam splitters [8], single-photon switching [9], diodes [10, 11], and photon transistor [12, 13], have been proposed.

As an important ingredients in building optical quantum network, optic switch has attracted more and more attentions. The research reveal that the atoms embedded in the low-dimensional system can act as an optic switch, and the photon propagation is controlled by the scattering process. According to its physical mechanism, these studies can be divided into two categories: (1) One-dimensions optic waveguide. One-dimensional continuum waveguide naturally arises in nanophotonic systems and the properties of photon transportation via waveguide has been extensively investigated [9–11, 14–21]. The commonality in these studies is that the dispersion relation in waveguide is linear, which is valid when the frequency of incident optic field is far away from the cut-off frequency of the waveguide. (2) Coupled-resonator waveguide (CRW) [22, 23]. CRW is a new and alternative candidate for building optical devices, and it can be realized by using coupled-cavity array [24], coupled superconducting transmission line resonators [25], or defect resonators in photonic crystals [26]. The nonlinear dispersion in CRW can result in many interesting phenomena, including single-photon quasibound states [27–29] and photon-atom bound state [30, 31]. In addition, such nonlinear dispersion is also helpful in designing the quantum switch for single photon of a specific frequency [32]. The single-frequency optic switch can be further generalized to wideband photon switch by placing many atoms individually in the resonators in a finite coordinate region of the waveguide [33]. Moreover, the scattering properties influenced by the non-rotating-wave-approximation term of Jaynes-Cummings model [34] and nonlocal coupling to the CRW [35] have also been analyzed.

Up till now, nearly all the schemes for optic switch are polarization-independent, meaning that the propagation of photon with identical polarization is controllable. In order to manipulate the polarization degree of freedom [8, 36, 37], a photon switch with the capability of polarization transformation is necessary. Recently, in a 1D waveguide, a kind of polarization-dependent photon switch has been studied by Tsoi and Law [17]. Employing this photon switch, an incident photon with an unknown polarization can be converted into a specified polarization, and the transmission probability is higher than that in polarizers obeying Malus’s law. As the CRW is one type of photonic system for quantum information processing, so it is natural to ask what the scattering property is in the CRW involving Λ-type three-level atoms, and how to optimize the performance of polarization-dependent photon switch. Here, we focus on these questions and provide some results which might be useful for designing optical quantum devices.

In this paper, we study the system of a coupled-resonator waveguide with *N _{a}* Λ-type three-level atoms. The analytic expressions of the eigenvectors of the Hamiltonian are derived by using discrete-coordinate scattering approach. Precisely, our discussions are divided into two scenarios:

*N*= 1 and

_{a}*N*> 1. We then give an analysis of the effect of system dissipation on the transfer spectra. Next, we provide numerical evaluations of our photon switching scheme, with the fidelity as the figure of merit for polarization transformation. Other influence of atomic decay and loss in resonators are also included. This numerical study further elucidates the physics mechanism under which the three-level atom could enable the control of the scattering and polarization of a single photon.

_{a}## 2. Model Hamiltonian and eigenvectors

We firstly introduce the system of CRW with Λ-type three-level atoms. As is shown in Fig. 1, the CRW is constructed by an array of coupled resonators. Due to the overlap of light modes of adjacent resonators, the photon can hop between the nearest-neighbor resonators with a hopping constant *ξ* [24]. There are *N _{a}* atoms individually embedded in one of the resonators in CRW. Here, the Λ-type atom in the

*j*th resonator consists of two degenerate ground states, which are represented as |

*H*〉

*and |*

_{j}*V*〉

*, and an excited state |*

_{j}*e*〉

*. These two ground states are coupled to the excited state with two orthogonal polarizations of the field,*

_{j}*H*and

*V*, with coupling strengths

*g*and

_{H}*g*respectively. The whole system Hamiltonian $H={H}_{\text{CRW}}+{\sum}_{j=1}^{{N}_{a}}{H}_{I,j}$ consists of two parts. The first part describes the free Hamiltonian of the CRW

_{V}*a*and ${a}_{j,s}^{\u2020}$ are the annihilation and creation operators associated with the polarization

_{j,s}*s*(

*s*=

*V*,

*H*) of the

*j*th two-mode resonator with frequency

*ω*, Ω is the difference in the energy level between excited state and ground state. |

*m*〉

*〈*

_{j,j}*n*| (

*m*,

*n*=

*e*,

*s*) is the dipole transition operator between |

*m*〉

*and |*

_{j}*n*〉

*. In a single resonator, the scattering of a single photon packet from a Λ-type three-level atom have been investigated by Chen*

_{j}*et al*[38].

In the following, by using discrete-coordinate scattering approach [27, 32, 33], we will analyze the scattering properties of a single photon in the CRW. We first label the resonators of CRW from −*N* to *N*, while the resonators with atoms are marked from 1 to *N _{a}*. Notice that the total excitation number operator
$\widehat{N}={\sum}_{j,s}{a}_{j,s}^{\u2020}{a}_{j,s}+{\sum}_{j}{|e\u3009}_{j,j}$ 〈

*e*| is a conserved observable, i.e., [

*N̂*,

*H*] = 0. So we restrict our analysis of photon scattering on the single-excitation subspace. We assume that a

*H*-polarized photon with eigenenergy

*E*incidents from the left, and all atoms are initially in the ground states |

*H*〉. To study the 1D single-photon elastic scattering problem governed by the total Hamiltonian

*H*, it needs to find the eigenstate of

*H*with eigenenergy

*E*for the incident photon through eigenvalue equation

*H*|

*E*〉 =

*E*|

*E*〉.

#### 2.1. *N*_{a} = 1 case

_{a}

For the most simple case, only one Λ-type three-level atom is embedded in the resonator 1, and the rest resonators are empty. Different from [32], a *V* -polarized photon may be released after the incident *H*-polarized photon is absorbed by the atom. We assume the stationary state of the system is

*j*〉 ⊗ |

_{s}*s*〉

_{1}denotes the state that the atom is at the ground state |

*s*〉 associated with a

*s*-polarized photon emitted into a mode of the

*j*th resonator, |0〉 ⊗ |

*e*〉

_{1}represents the state without any photon in the CRW and the atom is promoted to the excited state, ${u}_{j}^{s}$ and ${u}_{1}^{e}$ are the probability amplitudes for these two states respectively. Eq. (3) presents a complete set of stationary states of the total system for single-photon processed. Combining the eigenvalue equation, the bosonic commutation relations, and discrete coordinate representation, we derive the scattering equations

*j*≠ 1 are

*t*and

_{s}*r*for

_{s}*s*-polarized photon, respectively. Substituting these solutions into Eq. (4) in the region of resonators without atom, we can easily get the cosine-type dispersion for the incident photon with momentum

*k*. Next, considering the continuous conditions ${u}_{{1}^{+}}^{s}={u}_{{1}^{-}}^{s}$, we have Then from scattering equation (4) for

*j*= 1, we have

*s*-polarized photon

*E*− Ω is the energy detuning between atom and incident photon. Because the momentum of incident photon is real, the range of the detuning is

*ω*− Ω −2|

*ξ*| ⩽ Δ ⩽

*ω*−Ω + 2|

*ξ*|. It is easy to check that |

*t*|

_{H}^{2}+|

*r*|

_{H}^{2}+ |

*t*|

_{V}^{2}+ |

*r*|

_{V}^{2}= 1, which satisfies the conservation of probability in the scattering problem.

Since the dipole couplings for the two orthogonal polarizations are different, we can learn the photon transport properties by modulating the coupling strengths *g _{H}* and

*g*. When

_{V}*g*= 0, the |

_{V}*V*〉

_{1}is decoupled with the field of resonator. In this case, we can recover the result of control-lable scattering of a single

*H*-polarized photon in [32]. At the resonance, the

*H*-polarized photon is completely reflected. When

*g*≠ 0, after the atom is excited by absorbing incident photon, it can emit a

_{V}*V*-polarized photon with certain probability. From Eq. (16), we can conclude that the transmission coefficient |

*t*|

_{V}^{2}and reflection coefficient |

*r*|

_{V}^{2}of

*V*-polarized photon are the same. This phenomenon results from the fact that the released

*V*-polarized photon will be no preference of propagation direction. Especially, for the case of

*g*=

_{H}*g*=

_{V}*g*, the probabilities of emitting a polarized photon with forward and backward transmission are equal. However, due to the interfere with the incident

*H*-polarized photon, the transmission coefficient |

*t*|

_{H}^{2}is quite different from others. In Figs. 2(a)–2(d), we plot the transmission and reflection spectra for both of the polarization versus the detuning for only one atom. The parameters in these figures are chosen as

*g*=

_{H}*g*= 1,

_{V}*ω*= 5, Ω = 6, and

*ξ*= −1. These figures clearly show that |

*r*|

_{H}^{2}, |

*t*|

_{V}^{2}, and |

*r*|

_{V}^{2}are identical as we have pointed out. At the resonance, the transmission coefficient of

*H*-polarized photon |

*t*|

_{H}^{2}reaches its minimum and equals to the maximum of the other three transmission and reflection coefficients.

#### 2.2. *N*_{a} > 1 case

_{a}

When more atoms participate in the scattering processes, a more complicated set of eigenvector is needed. If all atoms are initially in the ground states |*H*〉, for a *H*-polarized photon traveled from left, the eigenstate of the total Hamiltonian *H* with eigenenergy *E* can be written in the following form

*j*to be ranging from −

*N*to

*N*, and

*l*is used to number the resonator with an atom, |

*j*〉 ⊗ |

_{H}*H′*〉 represents that

*H*-polarized photon locates in the

*j*th resonator and each of the three-level atoms is in the state |

*H*〉, |

*j*〉 ⊗ |

_{V}*V′*〉 denotes that the atom in the

_{l}*l*th resonator is transferred to state |

*V*〉 and releases a

*V*-polarized photon, which has transmitted to the

*j*th resonator, |0〉 ⊗ |

*e′*〉 represents that the single photon is absorbed and the atom in the

*l*th resonator is excited to state |

*e*〉. ${u}_{j}^{H}$, ${u}_{j,l}^{V}$, and ${u}_{l}^{e}$ are the corresponding amplitudes, respectively. Since all atoms are initially prepared in |

*H*〉 states, ${u}_{j,l}^{V}$, the amplitude of

*V*-polarized photon, is quite different from that of the single-atom case. In fact, once an atom is transferred to |

*V*〉 state, the scattered photon will propagate along the CRW without further interaction with other atoms. From the Schrödinger equation

*H*|

*E*〉 =

*E*|

*E*〉, the scattering equations for a single photon with discrete coordinate representation are given as

*V*-polarized photon may transfer into other resonators, the atom inside which is definitely in its |

*H*〉 state as we assumed. In this situation, the circumstance around the

*V*-polarized photon has no difference from the empty resonators. Substituting Eq. (21) into Eqs. (19) and (20), we get

*t*,

_{H}*r*,

_{H}*r*, and

_{V,l}*t*, respectively. Where

_{V,l}*k′*is the momentum of

*H*-polarized photon, which transmits in the resonators with an atom. Next we apply the boundary conditions for

*V*-polarized photon of the solutions, which give

*r*

_{V,l}e^{−}

*=*

^{ikl}*t*. Combining the cosine-type dispersion of the CRW and conservation of energy, we can obtain momentum

_{V,l}e^{ikl}*k′*, which is the solution of the transcendental equation

*s*-polarized photon are obtained as

In the limit of *N _{a}* → 1, we can get exactly the same results as shown in Fig. 2. While for

*g*= 0, the result in [33] can be recovered. In this case, we can realize perfect reflection of

_{V}*H*-polarized photon in a wide band of frequency.

Based on the above result, the polarization-dependent photon switch in CRW with multiple three-level atoms can be studied. In Fig. 3, we illustrate the transmission and reflection spectra of the two orthogonal polarized photons with different numbers of atoms. The other system parameters are the same as that in Fig. 2. For convenience, we write |*t _{H}*|

^{2}, |

*r*|

_{H}^{2}, ${\sum}_{l=1}^{{N}_{a}}{\left|{t}_{V,l}\right|}^{2}$, and ${\sum}_{l=1}^{{N}_{a}}{\left|{r}_{V,l}\right|}^{2}$ as

*T*,

_{H}*R*,

_{H}*T*, and

_{V}*R*, respectively. In Fig. 3(a), we see that there is an obvious drop for

_{V}*T*near the resonance when

_{H}*N*= 2. When more atoms are embedded into the system, the transmission coefficient of

_{a}*H*-polarized photon decrease quickly. When

*N*= 8, a band of forbidden transmission for incident photon, i.e.,

_{a}*T*≃ 0, emerges around the resonance. Such a band-gap-like structure for incident photon is also observed in a 1D waveguide [17]. The blocked photon is reflected with the same polarization, or convert into the vertical polarization. The probability of the former case is the reflection coefficient

_{H}*R*, which does not have distinct change except for the two boundaries of Δ as

_{H}*N*increases [Fig. 3(c)]. Meanwhile, due to the interference from multiple atoms,

_{a}*R*is much different from

_{H}*T*and

_{V}*R*even under the condition

_{V}*g*=

_{H}*g*. Because the released

_{V}*V*-polarized photon cannot transfer back to the

*H*-polarized photon, the transmission coefficient

*T*, which equals to the reflection coefficient

_{V}*R*, gradually accumulates as the number of atoms increases [Figs. 3(b) and 3(d)].

_{V}## 3. Influence of dissipation

For the ideal condition, there exists no dissipation during the photon scattering process. However, in realistic physical devices, the atomic decay and the resonator loss are unavoidable. The influence of dissipation on the photon transport characteristic needs to be investigated. Here we assume that the decay rate for atom is *γ _{a}*, and the two orthogonal polarized photons have the same decay rate

*γ*. In the calculation of transfer spectra, the imaginary parts −

_{c}*iγ*and −

_{a}*iγ*are included in the atomic frequency Ω and the coupled-resonator frequency

_{c}*ω*, respectively. Combining the definition of

*ω*(

*E*), which is now rewritten as $\omega \left({E}_{L}\right)=\frac{1}{\mathrm{\Delta}+i\left({\gamma}_{a}-{\gamma}_{c}\right)}$[33], and the

*k′*satisfies the relation

_{L}In Fig. 4, we plot the transmission and reflection spectra and photon current in the CRW embedded two atoms. The other system parameters are chosen as *g _{H}* =

*g*= 1,

_{V}*ω*= 5, Ω = 6, and

*ξ*= −1. For

*γ*= 0.1 and

_{c}*γ*= 0.2. The Figs. 4(a)–4(d) show that the transmission and reflection spectra has a little change compared with that in ideal case when the dissipation is not very large. While

_{a}*γ*increases to 0.4, obvious changes emerge near the resonance, especially for the transmission and reflection spectra of

_{a}*V*-polarized photon. This phenomenon implies that the effect of dissipation on polarization-dependent photon switch is not very large. Even under strong dissipation, the influence is limited in a certain range of frequency of incident photon. In Fig. 4(e), it displays that the photon current is conserved in the ideal case, while it is not conserved any more when the dissipation is included in the system. Meanwhile, the loss of photon current raises as the decay rate increases.

## 4. Polarization dependent transmission and reflection

Different from the switch of single photon with the same polarization [27–29,32–35], the photon switch discussed in this paper owns the ability of polarization transformation. In this section, we examine the performance of polarization transformation in a range of system parameters. To qualify such ability, we introduce the fidelity given by [17]

which presents the ratio of transmission photon with vertical polarization to total transmission photon. When*ℱ*= 0, the transmission photon is only

*H*-polarized, while

*ℱ*= 1 means that the transmission photon is completely transformed into a

*V*-polarized one.

We firstly study the effect of the number of atom in the CRW. As discussed above, the more atoms are embedded, the higher *T _{V}* and smaller

*T*are obtained [Figs. 3(a) and 3(b)]. According to the definition of

_{H}*ℱ*, it results in the increase of the fidelity. In Fig. 5, we present results of the fidelity (the red lines) as a function of detuning with multiple atoms. The other parameters are the same as that in Figure 2. It is shown that |

*t*|

_{H}^{2}= |

*t*|

_{V}^{2}at Δ = 0 in Fig. 2. For one atom in the CRW, a fidelity of

*ℱ*= 0.5 is thus obtained at resonance [Fig. 5(a)]. After embedding another atom into the CRW, we get a fidelity up to 0.8919 near the resonance point Δ = −0.0694 [Fig. 5(b)]. The highest fidelity at the resonance point raises as the number of atoms increases. When we embed further more atoms into the CRW, the perfect transformation of polarization appears in a wide band [Figs. 5(c)–5(f)]. It displays that for a given Ω the incident photon with higher energy are more likely to transmit through the CRW with converting its polarization. For a more realistic situation which includes the atomic decay and the resonator loss, we plot the fidelity spectrum

*ℱ*in Fig. 5 (the blue dashed line) with

*γ*= 0.3 and

_{a}*γ*= 0.15. It is shown in Fig. 5 that the frequency profiles of

_{c}*ℱ*are not sensitive to the system dissipation. In Figs. 5(d)–5(f), fidelity with decay even can be slightly higher than that in non-dissipative case (red line) in some range. This phenomenon results from the facts that input photon with

*H*polarization is almost completely blocked and loss when

*N*is very large [Fig. 6(a)], and on the other hand, the transmission

*T*of

_{V}*V*-polarized photon asymptotically closes to be a certain value even in the dissipation case when

*N*> 15 [Fig. 6(b)]. Figs. 6(a) and 6(b) show that the dissipation reduces the transmissions for both of the polarized photon. However, when differences

*δ*between the transmissions

_{s}*T*and

_{s}*T*satisfy the relation

_{s}L*T*>

_{V}δ_{H}*T*, the fidelity with dissipation will be larger than that without dissipation [Fig. 6(c)]. A similar phenomenon is also revealed in 1D waveguide [17] even when the decay rate is comparable with the atomic frequency.

_{H}δ_{V}Next, we discuss the influence of other system parameters on fidelity, such as the detuning Δ and the dipole couplings *g _{V}*,

*g*with two orthogonal polarizations. We plot the contour map of the fidelity in Fig. 7. The contour map can be regarded as a kind of phase diagram. In Fig. 7, the white areas denote that the

_{H}*ℱ*is near one, while the dark areas indicate that the

*ℱ*is close to zero. In Fig. 7(a), we choose the same parameters as that in Fig. 5 and

*g*= 1. It shows that the bigger

_{H}*g*, the wider range of white areas. However, the white areas appear due to different mechanism. In the strong coupling regime, the bigger

_{V}*g*means that the excited state of atom is easier to release a

_{V}*V*-polarized photon, which leads to

*ℱ*= 1. On the other hand, when

*g*is in the weak coupling regime, the

_{V}*ℱ*= 1 results from the band of forbidden transmission for incident photon, i.e.,

*T*≃ 0 [Fig. 3(a)]. In Fig. 7(b), we set

_{H}*g*= 1. It shows that associated with the increasing of

_{V}*g*the white areas are enlarged. Together with Figs. 7(a) and 7(b), it seems that fidelity becomes bigger accompany with the growing of both dipole couplings. This is further demonstrated in Fig. 8 when the contour map of the fidelity

_{H}*ℱ*is plotted as a function of the diploe couplings

*g*and

_{V}*g*for

_{H}*N*= 8 and Δ = −1.

_{a}## 5. Conclusion

To conclude, the scattering properties of a 1D CRW with *N _{a}* Λ-type atoms have been investigated. Based on the discrete-coordinate scattering method, we have derived analytic expressions of transmission and reflection coefficients for the case of a single atom. Then we generalize it to the case of multiple atoms. For the incident

*H*-polarized photon, we have shown that a band of forbidden transmission for incident photon emerges near the resonance as the number of atoms increases. Meanwhile, the probability of converting the

*H*-polarized photon to the

*V*-polarized photon raises. Such

*V*-polarized photon propagates forward and backward equally. We then prove the transmission and reflection spectra are not sensitive to the atomic decay and the resonator loss, although such dissipations of system can reduce the photon current in the CRW. Furthermore, we introduce fidelity to qualify the ability of polarization transformation, and study the fidelity in varieties of system parameters, such as number of atoms, energy detuning, dipole couplings with orthogonal polarizations. To get a better fidelity, we should embed more atoms into the CRW, and keep higher dipole couplings. The results of these analyses may be helpful for designing a polarization-dependent photon switch, which can blockade and convert the incident photon with certain polarization.

## Acknowledgments

Y.L.D thanks Dr. Yan-Xiao Gong for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grants No. 11074184, No. 11204197, and No. 11204379), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20103201120002), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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